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1    NORMED ALGEBRAS       305

spectrum and Gelfand transformation (15.3), have found many other applica-
tions in modern analysis, notably in the theory of analytic functions. Some of
these applications are touched on in the problems, and the reader interested in
this aspect is referred to [35] and [29].

The central part of this chapter is the study of representations of algebras
with involution, which enables such an algebra, given "abstractly," to be
"realized" as an algebra of operators on a Hilbert space.

The essential notion in the modern development of this theory is that of a
Hilbert form, which is closely related to that of a Hilbert algebra (15.7). We
shall study in detail only two particular aspects of the theory of Hilbert
algebras: the first (15.8) prepares the way for the theory of representations of
compact groups (Chapter XXI), and the second (15.9) for spectral theory and
harmonic analysis (Chapter XXII). The reader who wishes to go further
(notably in view of the deep and difficult theory of representations of locally
compact groups) is warmly recommended to read the two beautiful volumes
by J. Dixmier ([24] and [25]) which dominate the subject.

The Hilbert spectral theory (15.10 and 15.11) appears in our treatment as
an immediate particular case of the general theorem of Bochner-Godement
(15.9). It can be reached more directly and rapidly (Section 15.10, Problem 2)
from the Gelfand-Neumark theorem (15.4), but it seemed to us to be more
instructive to deduce it from a much more powerful theorem which is the
cornerstone of harmonic analysis, even if this requires a small additional
effort.

The applications of spectral theory are not limited to those referred to
above. Among the most celebrated we should mention at least the following:

(1)  one of the most elegant theories in Analysis, namely the "moment
problem" inaugurated by Stieltjes, with its many ramifications (analytic
functions, orthogonal polynomials, Jacobi matrices, continued fractions, etc),
which fits admirably into the theory of unbounded Hermitian operators;

(2)  the interesting relations between ergodic theory and spectral theory;

(3) perturbation theory. Some of the important results of these theories are
mentioned in the problems, and the reader is referred for more ample informa-
tion to the works [20], [28], [30], and [32] in the References.

1.  NORMED ALGEBRAS

When we speak of algebras in this chapter, we shall always mean algebras
over the field of complex numbers C. A normed algebra is defined to be an
algebra A endowed with a norm JCH* \\x\\ (5.1) satisfying the inequality

(15.1.1)                                        \\xy\\ ^ \\x\\  \\y\\

for all x, y in A.how that as y